Numerical Solution Of Two-dimensional Sine-Gordon Equation Using The Predictor-corrector Method

In this paper, A family of finite difference method are presented for solving two-dimensional nonlinear sine-Gordon equation with the initial/boundary-value problem. The local truncation error, stability and convergence of the method are analyzed. First of all, based on rational approximation, we construct a three-layer of implicit difference scheme, the format contains nonlinear term sin (u). To obtain the numerical solutions on every layer, we need to solve a large nonlinear differential equation. In order to avoid solving the nonlinear system, a predictor-corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Using the methods of lines, we propose three different approximations to nonlinear term sin (u), so we get three different layer explicit schemes as predictors. The numerical cases are given in the end, including the most known from the bibliography line and ring solitons and the results show that this method is effective and reliable.